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monoidal model category

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Monoidal categories

Contents

Idea

A monoidal model category is a model category which is also a closed monoidal category in a compatible way. In particular, its homotopy category inherits a closed monoidal structure.

Definition

Definition

A (symmetric) monoidal model category is model category 𝒞\mathcal{C} equipped with the structure of a closed symmetric monoidal category (𝒞,,I)(\mathcal{C}, \otimes, I) such that the following two compatibility conditions are satisfied

  1. (pushout-product axiom) For every pair of cofibrations f:XYf \colon X \to Y and f:XYf' \colon X' \to Y', their pushout-product, hence the induced morphism out of the cofibered coproduct over ways of forming the tensor product of these objects

    (XY) XX(YX)YY, (X \otimes Y') \coprod_{X \otimes X'} (Y \otimes X') \longrightarrow Y \otimes Y' \,,

    is itself a cofibration, which, furthermore, is acyclic if ff or ff' is.

    (Equivalently this says that the tensor product :C×CC\otimes : C \times C \to C is a left Quillen bifunctor.)

  2. (unit axiom) For every cofibrant object XX and every cofibrant resolution QIp I*\emptyset \hookrightarrow Q I \stackrel{p_I}{\longrightarrow} \ast of the tensor unit II, the resulting morphism

    QIXp IXIXX Q I \otimes X \stackrel{p_I \otimes X}{\longrightarrow} I\otimes X \stackrel{\simeq}{\longrightarrow} X

is a weak equivalence.

Remark

The pushout-product axiom in def. 1 implies that for XX a cofibrant object, then the functor X()X \otimes (-) is preserves cofibrations and acyclic cofibrations.

In particular if the tensor unit II happens to be cofibrant, then the unit axiom in def. 1 is implied by the pushout-product axiom. (Because then QIIQ I \to I is a weak equivalence between cofibrant objects and such are preserved by functors that preserve acyclic cofibrations, by Ken Brown's lemma. )

Definition

We say a monoidal model category, def. 1, satisfies the monoid axiom, def. 1, if every morphism that is obtained as a transfinite composition of pushouts of tensor products XfX\otimes f of acyclic cofibrations ff with any object XX is a weak equivalence.

(Schwede-Shipley 00, def. 3.3.).

Remark

In particular, the axiom in def. 2 says that for every object XX the functor X()X \otimes (-) sends acyclic cofibrations to weak equivalences.

Properties

Monoidal homotopy category

Proposition

Let (𝒞,,I)(\mathcal{C}, \otimes, I) be a monoidal model category. Then the left derived functor of the tensor product exists and makes the homotopy category into a monoidal category (Ho(𝒞), L,γ(I))(Ho(\mathcal{C}), \otimes^L, \gamma(I)).

If in in addition (𝒞,)(\mathcal{C}, \otimes) satisfies the monoid axiom, then the localization functor γ:𝒞Ho(𝒞)\gamma\colon \mathcal{C}\to Ho(\mathcal{C}) carries the structure of a lax monoidal functor

γ:(𝒞,,I)(Ho(𝒞), L,γ(I)). \gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.
Proof

Consider the explicit model of Ho(𝒞)Ho(\mathcal{C}) as the category of fibrant-cofibrant objects in 𝒞\mathcal{C} with left/right-homotopy classes of morphisms between them (discussed at homotopy category of a model category).

A derived functor exists if its restriction to this subcategory preserves weak equivalences. Now the pushout-product axiom implies that on the subcategory of cofibrant objects the functor \otimes preserves acyclic cofibrations, and then the preservation of all weak equivalences follows by Ken Brown's lemma.

Hence L\otimes^L exists and its associativity follows simply by restriction. It remains to see its unitality.

To that end, consider the construction of the localization functor γ\gamma via a fixed but arbitrary choice of (co-)fibrant replacements QQ and RR, assumed to be the identity on (co-)fibrant objects. We fix notation as follows:

Cofi XQXWFibp xX,XWCofj XRXFibq x*. \emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X \;\;\,,\;\; X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast \,.

Now to see that γ(I)\gamma(I) is the tensor unit for L\otimes^L, notice that in the zig-zag

(RQI)(RQX)j QI(RQX)(QI)(RQX)(QI)j QX(QI)(QX)p I(QX)IQXQX (R Q I) \otimes (R Q X) \overset{j_{Q I} \otimes (R Q X)}{\longleftarrow} (Q I) \otimes (R Q X) \overset{(Q I)\otimes j_{Q X}}{\longleftarrow} (Q I) \otimes (Q X) \overset{p_I \otimes (Q X)}{\longrightarrow} I \otimes Q X \simeq Q X

all morphisms are weak equivalences: For the first two this is due to the pushout-product axiom, for the third this is due to the unit axiom on a monoidal model category. It follows that under γ()\gamma(-) this zig-zig gives an isomorphism

γ(I) Lγ(X)γ(X) \gamma(I) \otimes^L \gamma(X)\simeq \gamma(X)

and similarly for tensoring with γ(I)\gamma(I) from the right.

To exhibit lax monoidal structure on γ\gamma, we need to construct a natural transformation

γ(X) Lγ(Y)γ(XY) \gamma(X) \otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y)

and show that it satisfies the the appropriate associativity and unitality condition.

By the definitions at homotopy category of a model category, the morphism in question is to be of the form

(RQX)(RQY)RQ(XY) (R Q X) \otimes (R Q Y) \longrightarrow R Q (X\otimes Y)

To this end, consider the zig-zag

(RQX)(RQY)CofWj QXRQY(QX)(RQY)CofW(QX)j QY(QX)(QY)p X(QY)X(QY)Yp YXY, (R Q X) \otimes (R Q Y) \underoverset{\in Cof \cap W}{j_{Q X} \otimes R Q Y}{\longleftarrow} (Q X) \otimes (R Q Y) \underoverset{\in Cof \cap W}{(Q X) \otimes j_{Q Y} }{\longleftarrow} (Q X) \otimes (Q Y) \overset{p_X \otimes (Q Y)}{\longrightarrow} X \otimes (Q Y) \overset{Y \otimes p_Y}{\longrightarrow} X \otimes Y \,,

and observe that the two morphisms on the left are weak equivalences, as indicated, by the pushout-product axiom satisfied by \otimes.

Hence applying γ\gamma to this zig-zag, which is given by the two horizontal part of the following digram

(RQX)(RQY) R(QXQY) RQ(XY) id j QXQY j Q(XY) id id p XY (RQX)(RQY) CofWj QXj QY (QX)(QY) p Xp Y XY, \array{ (R Q X) \otimes (R Q Y) &\longleftarrow& R( Q X \otimes Q Y ) &\longrightarrow& R Q (X \otimes Y) \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y}}} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}}} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{X\otimes Y}}} \\ (R Q X) \otimes (R Q Y) &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y}}{\longleftarrow}& (Q X) \otimes (Q Y) &\overset{p_X \otimes p_Y}{\longrightarrow}& X \otimes Y } \,,

and inverting the first two morphisms, this yields a natural transformation as required.

To see that this satisfies associativity if the monoid axiom holds, tensor the entire diagram above on the right with (RQZ)(R Q Z) and consider the following pasting composite:

(RQX)(RQY)(RQZ) R(QXQY)(RQZ) (RQ(XY))(RQZ) id j QXQYid j Q(XY)id Q(XY)(RQZ) idj QZ Q(XY)(QZ) id id p (XY)id () p (XY)id (RQX)(RQY)(RQZ) CofWj QXj QYid (QX)(QY)(RQZ) p Xp Yid XY(RQZ) idj QZ XYQZ idp Z XYZ, \array{ (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\longleftarrow& R( Q X \otimes Q Y ) \otimes (R Q Z) &\longrightarrow& (R Q (X \otimes Y)) \otimes (R Q Z) \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y} \otimes id }} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}\otimes id }} \\ && && Q(X \otimes Y) \otimes (R Q Z) &\overset{id \otimes j_{Q Z}}{\longleftarrow}& Q(X\otimes Y) \otimes (Q Z) \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{(X\otimes Y)} \otimes id }} &(\star)& \downarrow^{\mathrlap{p_{(X \otimes Y)} \otimes id}} \\ (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y} \otimes id}{\longleftarrow}& (Q X) \otimes (Q Y) \otimes (R Q Z) &\overset{p_X \otimes p_Y \otimes id}{\longrightarrow}& X \otimes Y \otimes (R Q Z) &\underset{id \otimes j_{Q Z}}{\longleftarrow}& X\otimes Y \otimes Q Z &\overset{id \otimes p_Z}{\longrightarrow}& X \otimes Y \otimes Z } \,,

Observe that under γ\gamma the total top zig-zag in this diagram gives

(γ(X) Lγ(Y)) Lγ(Z)γ(XY) Lγ(Z). (\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) \to \gamma(X\otimes Y)\otimes^L \gamma(Z) \,.

Now by the monoid axiom (but not by the pushout-product axiom!), the horizontal maps in the square in the bottom right (labeled \star) are weak equivalences. This implies that the total horizontal part of the diagram is a zig-zag in the first place, and that under γ\gamma the total top zig-zag is equal to the image of that total bottom zig-zag. But by functoriality of \otimes, that image of the bottom zig-zag is

γ(p Xp Yp Z)γ(j QXj QYj QZ) 1. \gamma(p_X \otimes p_Y \otimes p_Z) \circ \gamma(j_{Q X} \otimes j_{Q Y} \otimes j_{Q Z})^{-1} \,.

The same argument applies to left tensoring with RQZR Q Z instead of right tensoring, and so in both cases we reduce to the same morphism in the homotopy category, thus showing the associativity condition on the transformation that exhibits γ\gamma as a lax monoidal functor.

Examples

Classical homotopy theory

Homological algebra and stable homotopy theory

With respect to a symmetric monoidal smash product of spectra:

(Schwede-Shipley 00, MMSS 00, theorem 12.1 (iii) with prop. 12.3)

The standard example of a monoidal model category whose unit is not cofibrant is the category of EKMM S-modules.

Categorical model structures

Model structure on GG-objects

Assumption

Let \mathcal{E} be a category equipped with the structure of

such that

Proposition

Under these conditions there is for each finite group GG the structure of a monoidal model category on the category BG\mathcal{E}^{\mathbf{B}G} of objects in \mathcal{E} equipped with a GG-action, for which the forgetful functor

BG \mathcal{E}^{\mathbf{B}G} \to \mathcal{E}

preserves and reflects fibrations and weak equivalences.

See for instance (BergerMoerdijk 2.5).

Model structure on monoids

See model structure on monoids in a monoidal model category.

References

A general standard reference is

The monoidal structures for a symmetric monoidal smash product of spectra are due to

The monoidal model structure on BG\mathcal{E}^{\mathbf{B}G} is discussed for instance in

Relation to symmetric monoidal (infinity,1)-categories is discussed in

Revised on May 2, 2016 05:01:51 by Urs Schreiber (131.220.184.222)